Finite Group Databasebeta

← #12126729

Subgroups — #12126729

Order 1280.

1148 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3458C211yesmaximal, characteristic, complemented
2C2×#65295C211yesmaximal, characteristic
3C2×#65294C211yesmaximal, characteristic
4C2×#65294C211yesmaximal, characteristic
5#10602635C218yesmaximal, complemented
6C2×#64753C211yesmaximal, characteristic, complemented
7#10602371C211yesmaximal, characteristic, complemented
8C2×#64751C211yesmaximal, characteristic, complemented
9C22×#709C2211yescharacteristic
10C5×#544—22no—
11C5×#544—22no—
12C5×#544—22no—
13C5×#543—22no—
14C5×#533—22no—
15C5×#533—22no—
16C5×#533C2214yescomplemented
17C2×C5×#117C2211yescharacteristic
18C2×C5×#117C2211yescharacteristic
19C2×C5×#117C2211yescharacteristic
20C2×#4072C2211yescharacteristic
21C22×#681C2211yescharacteristic
22C22×#681C2211yescharacteristic
23#65295—22no—
24#65295C2214yes—
25#65294—22no—
26#65294—22no—
27#65294—22no—
28#65294C2214yes—
29C2×#3938—22no—
30C2×#3937—22no—
31C2×#3937C2214yes—
32#64753C2214yescomplemented
33#64751C2214yescomplemented
34#64743C2214yescomplemented
35#23472—51nomaximal, Sylow
36C23×C4×C5C2311yescharacteristic
37C22×C22⋊D5—21no—
38C2×#709—21no—
39C2×#709—21no—
40C2×#709—21no—
41C2×#709—21no—
42C2×#709—21no—
43C2×#709—21no—
44C2×#709—21no—
45C2×#709—21no—
46C2×#709—21no—
47C2×#709—21no—
48C2×#709Q812yescomplemented
49C2×#709C2312yes—
50C22×D20Q811yescharacteristic, complemented
51C22×C4×D5—21no—
52C22×Dic10Q811yescharacteristic, complemented
53C5×C4⋊Q8—24no—
54C5×C4⋊Q8—24no—
55C5×C4⋊Q8—24no—
56C5×#125—24no—
57C5×#125—24no—
58C5×#125—24no—
59C5×C22⋊Q8—28no—
60C5×C22⋊Q8—28no—
61C5×C22⋊Q8—28no—
62C4×C5×Q8—28no—
63C5×#117—21no—
64C5×#117—21no—
65C5×#117—21no—
66C5×#117—21no—
67C5×#117—21no—
68C5×#117—21no—
69C5×#117D412yes—
70C5×#117D412yes—
71C5×#117D412yes—
72C5×#117D412yes—
73C5×#117C2312yes—
74C5×#117C2312yes—
75C2×C5×C4⋊C4—21no—
76C2×C5×C4⋊C4—21no—
77C2×C5×C4⋊C4—21no—
78C2×C5×C22⋊C4—21no—
79C2×C5×C22⋊C4—21no—
80C2×C5×C22⋊C4D412yes—
81C2×C42×C5—21no—
82C2×C42×C5D412yes—
83C2×C42×C5C2312yes—
84C2×C22⋊Dic5—21no—
85#4072—21no—
86#4072—21no—
87#4072—21no—
88#4072C2312yes—
89C2×C4⋊Dic5—21no—
90C2×C4×Dic5—21no—
91C2×#681—21no—
92C2×#681—21no—
93C2×#681—21no—
94C2×#681—21no—
95C2×#681—21no—
96C2×#681—21no—
97C2×#681—21no—
98C2×#681—21no—
99C2×#681—21no—
100C2×#681—21no—
101C2×#681C2312yes—
102C2×#681C2312yes—
103C22×#168—21no—
104C22×#168—21no—
105#3938—28no—
106#3937—28no—
107#3937—28no—
108#3458D511yescharacteristic, complemented
109C2×#421—51no—
110C2×#421—51no—
111C2×#410—51no—
112#2031—58no—
113C2×#344—51no—
114#1733—51no—
115C2×H(3,ℤ/4)—51no—
116C24×C5C2×D411yescharacteristic
117C23×D5—21no—
118C2×C5×Q8—28no—
119C22×C4×C5—22no—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5C2×D411yescharacteristic
123C22×C4×C5C2×D411yescharacteristic
124C22×C4×C5C2×D411yescharacteristic
125C22×C4×C5C2×D411yescharacteristic
126C22×C4×C5C2×D411yescharacteristic
127C22×C4×C5C2×Q811yescharacteristic
128C22×C4×C5C2×Q811yescharacteristic
129C22×C4×C5#4111yescharacteristic
130C22×C4×C5#4111yescharacteristic
131C22×C4×C5#4111yescharacteristic
132C22×C4×C5#4111yescharacteristic
133C22×C4×C5#4111yescharacteristic
134C22×C4×C5#4111yescharacteristic
135C22×C4×C5C2411yescharacteristic
136C2×C22⋊D5—22no—
137C2×C22⋊D5—22no—
138C2×C22⋊D5—41no—
139C2×C22⋊D5—41no—
140C2×C22⋊D5—41no—
141C2×C22⋊D5—41no—
142C22×Dic5—21no—
143C22×Dic5—41no—
144#709—22no—
145#709—22no—
146#709—22no—
147#709—22no—
148#709—22no—
149#709—22no—
150#709—22no—
151#709—22no—
152#709—41no—
153#709—41no—
154#709—41no—
155#709—41no—
156#709—41no—
157#709—41no—
158#709—41no—
159#709—41no—
160C2×D20—21no—
161C2×D20—21no—
162C2×D20—21no—
163C2×D20—21no—
164C2×D20—21no—
165C2×D20C2×Q812yes—
166C2×C4×D5—22no—
167C2×C4×D5—22no—
168C2×C4×D5—41no—
169C2×C4×D5—41no—
170C2×C4×D5—41no—
171C2×C4×D5—41no—
172C2×Dic10—21no—
173C2×Dic10—21no—
174C2×Dic10—21no—
175C2×Dic10—21no—
176C2×Dic10—21no—
177C2×Dic10C2×Q812yes—
178C5×C4⋊C4—28no—
179C5×C4⋊C4—28no—
180C5×C4⋊C4—28no—
181C5×C4⋊C4—22no—
182C5×C4⋊C4—22no—
183C5×C4⋊C4—22no—
184C5×C4⋊C4—22no—
185C5×C4⋊C4—22no—
186C5×C4⋊C4—22no—
187C5×C22⋊C4—22no—
188C5×C22⋊C4—22no—
189C5×C22⋊C4—22no—
190C5×C22⋊C4—22no—
191C5×C22⋊C4—22no—
192C5×C22⋊C4C2×D414yes—
193C42×C5—22no—
194C42×C5—22no—
195C42×C5—22no—
196C42×C5—22no—
197C42×C5C2×D414yes—
198C42×C5C2×D414yes—
199C22⋊Dic5—22no—
200C22⋊Dic5—41no—
201C4⋊Dic5—22no—
202C4⋊Dic5—41no—
203C4×Dic5—22no—
204C4×Dic5—41no—
205#681—22no—
206#681—22no—
207#681—22no—
208#681—22no—
209#681—41no—
210#681—41no—
211#681—41no—
212#681—41no—
213C2×#168—22no—
214C2×#168—22no—
215C2×#168—41no—
216C2×#168—41no—
217C2×#168—41no—
218C2×#168—41no—
219C22×#41—51no—
220C22×M16—51no—
221C22×M16—51no—
222#544—22no—
223#544—22no—
224#544—22no—
225#543—22no—
226#533—22no—
227#533—22no—
228#533C2×D514yescomplemented
229C2×#117—51no—
230C2×#117C2×D511yescharacteristic
231C2×#117C2×D511yescharacteristic
232C2×#117C2×D511yescharacteristic
233#421—54no—
234#421—102no—
235#421—102no—
236#421—102no—
237C2×#104—54no—
238C2×#104—102no—
239#410—54no—
240#410—102no—
241C2×#99—102no—
242#344—54no—
243#339—54no—
244H(3,ℤ/4)—54no—
245C23×C5—81no—
246C23×C5#12011yescharacteristic
247C23×C5#12011yescharacteristic
248C23×C5C22⋊Q811yescharacteristic
249C23×C5C22⋊Q811yescharacteristic
250C23×C5#12411yescharacteristic
251C23×C5#12411yescharacteristic
252C23×C5C22×D411yescharacteristic
253C22×D5—22no—
254C22×D5—41no—
255C22×D5—41no—
256C22×D5—82no—
257C5×Q8—88no—
258C2×C4×C5—24no—
259C2×C4×C5—24no—
260C2×C4×C5—24no—
261C2×C4×C5—82no—
262C2×C4×C5—82no—
263C2×C4×C5—82no—
264C2×C4×C5#12011yescharacteristic
265C2×C4×C5#12011yescharacteristic
266C2×C4×C5C22⋊Q811yescharacteristic
267C2×C4×C5C22⋊Q811yescharacteristic
268C2×C4×C5C22⋊Q811yescharacteristic
269C2×C4×C5C22⋊Q811yescharacteristic
270C2×C4×C5C22⋊Q811yescharacteristic
271C2×C4×C5C22⋊Q811yescharacteristic
272C2×C4×C5C22⋊Q811yescharacteristic
273C2×C4×C5C22⋊Q811yescharacteristic
274C2×C4×C5C22⋊Q811yescharacteristic
275C2×C4×C5C22⋊Q811yescharacteristic
276C2×C4×C5#12411yescharacteristic
277C2×C4×C5#12411yescharacteristic
278C2×C4×C5#12411yescharacteristic
279C2×C4×C5#12411yescharacteristic
280C2×C4×C5#12411yescharacteristic
281C2×C4×C5#12411yescharacteristic
282C2×C4×C5#12411yescharacteristic
283C2×C4×C5#12411yescharacteristic
284C2×C4×C5#12411yescharacteristic
285C2×C4×C5#12411yescharacteristic
286C2×C4×C5C22×D411yescharacteristic
287C2×C4×C5C22×D411yescharacteristic
288C2×C4×C5C22×Q811yescharacteristic
289C2×C4×C5C2×#4111yescharacteristic
290C2×C4×C5C2×#4111yescharacteristic
291C2×C4×C5C2×#4111yescharacteristic
292C22⋊D5—42no—
293C22⋊D5—42no—
294C22⋊D5—42no—
295C22⋊D5—42no—
296C2×Dic5—22no—
297C2×Dic5—41no—
298C2×Dic5—41no—
299C2×Dic5—42no—
300C2×Dic5—82no—
301D20—22no—
302D20—22no—
303D20—41no—
304D20—41no—
305C4×D5—42no—
306C4×D5—42no—
307C4×D5—42no—
308C4×D5—42no—
309Dic10—22no—
310Dic10—22no—
311Dic10—41no—
312Dic10—41no—
313#168—42no—
314#168—42no—
315C2×#41—52no—
316C2×#41—52no—
317C2×#41—101no—
318C2×#41—101no—
319C2×#41—101no—
320C2×#41—101no—
321C2×#41—101no—
322C2×#41—101no—
323C2×#41—101no—
324C2×#41—101no—
325C2×#41—101no—
326C2×#41—101no—
327C22×Q8—51no—
328C22×D4—51no—
329C22×D4—101no—
330C23×C4—101no—
331C23×C4C22×D511yescharacteristic
332C2×M16—52no—
333C2×M16—52no—
334C2×M16—101no—
335C2×M16—101no—
336C2×M16—101no—
337C2×M16—101no—
338C2×M16—101no—
339C2×M16—101no—
340C2×M16—101no—
341C2×M16—101no—
342C2×M16—101no—
343C2×M16—101no—
344C22×C8—101no—
345C22×C8—101no—
346C4⋊Q8—24no—
347C4⋊Q8—24no—
348C4⋊Q8—24no—
349#125—24no—
350#125—24no—
351#125—24no—
352C22⋊Q8—28no—
353C22⋊Q8—28no—
354C22⋊Q8—28no—
355C4×Q8—28no—
356#117—21no—
357#117—21no—
358#117—21no—
359#117—21no—
360#117—21no—
361#117—21no—
362#117—52no—
363#117—101no—
364#117—101no—
365#117—101no—
366#117C22⋊D512yes—
367#117C22⋊D512yes—
368#117C22⋊D512yes—
369#117C22⋊D512yes—
370#117C22×D512yes—
371#117C22×D512yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—101no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—101no—
379C2×C22⋊C4C22⋊D512yes—
380C2×C42—21no—
381C2×C42—101no—
382C2×C42C22⋊D512yes—
383C2×C42C22×D512yes—
384#104—108no—
385#104—108no—
386#99—108no—
387C22×C5—41no—
388C22×C5—41no—
389C22×C5—41no—
390C22×C5—41no—
391C22×C5—41no—
392C22×C5—41no—
393C22×C5—41no—
394C22×C5#39311yescharacteristic
395C22×C5#39311yescharacteristic
396C22×C5#39311yescharacteristic
397C22×C5#39311yescharacteristic
398C22×C5C2×#12011yescharacteristic
399C22×C5C2×C22⋊Q811yescharacteristic
400C22×C5C2×#12411yescharacteristic
401C2×D5—42no—
402C2×D5—82no—
403C2×D5—82no—
404C2×D5—82no—
405C4×C5—44no—
406C4×C5—44no—
407C4×C5—44no—
408C4×C5#39311yescharacteristic
409C4×C5#39311yescharacteristic
410C4×C5#39311yescharacteristic
411C4×C5#39311yescharacteristic
412C4×C5C2×C22⋊Q811yescharacteristic
413C4×C5C2×C22⋊Q811yescharacteristic
414C4×C5C2×#12411yescharacteristic
415C4×C5C2×#12411yescharacteristic
416Dic5—42no—
417Dic5—82no—
418C24—101no—
419C24C2×C22⋊D511yescharacteristic
420#41—102no—
421#41—102no—
422#41—102no—
423#41—102no—
424#41—102no—
425#41—102no—
426#41—102no—
427#41—102no—
428#41—201no—
429#41—201no—
430#41—201no—
431#41—201no—
432#41—201no—
433#41—201no—
434#41—201no—
435#41—201no—
436C2×Q8—28no—
437C2×Q8—52no—
438C2×Q8—101no—
439C2×Q8—101no—
440C2×Q8—101no—
441C2×Q8—101no—
442C2×Q8—101no—
443C2×D4—52no—
444C2×D4—102no—
445C2×D4—102no—
446C2×D4—101no—
447C2×D4—101no—
448C2×D4—101no—
449C2×D4—101no—
450C2×D4—101no—
451C2×D4—201no—
452C2×D4—201no—
453C2×D4—201no—
454C2×D4—201no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—101no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—201no—
462C22×C4—201no—
463C22×C4—201no—
464C22×C4—201no—
465C22×C4—201no—
466C22×C4Q8×D511yescharacteristic
467C22×C4Q8×D511yescharacteristic
468C22×C4#71311yescharacteristic
469C22×C4#71311yescharacteristic
470C22×C4#71311yescharacteristic
471C22×C4#71311yescharacteristic
472C22×C4#71311yescharacteristic
473C22×C4#71311yescharacteristic
474C22×C4C2×C22⋊D511yescharacteristic
475C22×C4C2×C22⋊D511yescharacteristic
476C22×C4C2×C22⋊D511yescharacteristic
477C22×C4C2×C22⋊D511yescharacteristic
478C22×C4C2×C22⋊D511yescharacteristic
479C22×C4C23×D511yescharacteristic
480M16—102no—
481M16—102no—
482M16—102no—
483M16—102no—
484M16—201no—
485M16—201no—
486M16—201no—
487M16—201no—
488C2×C8—102no—
489C2×C8—102no—
490C2×C8—201no—
491C2×C8—201no—
492C2×C8—201no—
493C2×C8—201no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—102no—
504C4⋊C4—201no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—102no—
511C22⋊C4—201no—
512C22⋊C4C2×C22⋊D514yes—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—102no—
518C42—201no—
519C42C2×C22⋊D514yes—
520C42C2×C22⋊D514yes—
521C2×C5—21no—
522C2×C5—21no—
523C2×C5—21no—
524C2×C5—21no—
525C2×C5—21no—
526C2×C5—21no—
527C2×C5—21no—
528C2×C5C2×#39311yescharacteristic
529D5—82no—
530C23—81no—
531C23—102no—
532C23—201no—
533C23—201no—
534C23—402no—
535C23#409211yescharacteristic
536C23#409211yescharacteristic
537C23#409311yescharacteristic
538C23#409311yescharacteristic
539C23#409911yescharacteristic
540C23#409911yescharacteristic
541C23C22×C22⋊D511yescharacteristic
542Q8—88no—
543Q8—102no—
544Q8—102no—
545Q8—201no—
546Q8—201no—
547D4—102no—
548D4—102no—
549D4—202no—
550D4—202no—
551D4—202no—
552D4—202no—
553D4—201no—
554D4—201no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—102no—
562C2×C4—201no—
563C2×C4—201no—
564C2×C4—202no—
565C2×C4—202no—
566C2×C4—202no—
567C2×C4—202no—
568C2×C4—202no—
569C2×C4—402no—
570C2×C4#409211yescharacteristic
571C2×C4#409211yescharacteristic
572C2×C4#409211yescharacteristic
573C2×C4#409211yescharacteristic
574C2×C4#409211yescharacteristic
575C2×C4#409211yescharacteristic
576C2×C4#409211yescharacteristic
577C2×C4#409211yescharacteristic
578C2×C4#409211yescharacteristic
579C2×C4#409211yescharacteristic
580C2×C4#409311yescharacteristic
581C2×C4#409311yescharacteristic
582C2×C4#409311yescharacteristic
583C2×C4#409311yescharacteristic
584C2×C4#409311yescharacteristic
585C2×C4#409311yescharacteristic
586C2×C4#409311yescharacteristic
587C2×C4#409311yescharacteristic
588C2×C4#409311yescharacteristic
589C2×C4#409311yescharacteristic
590C2×C4#409911yescharacteristic
591C2×C4#409911yescharacteristic
592C2×C4C2×Q8×D511yescharacteristic
593C2×C4C2×#71311yescharacteristic
594C2×C4C2×#71311yescharacteristic
595C2×C4C2×#71311yescharacteristic
596C2×C4C22×C22⋊D511yescharacteristic
597C2×C4C22×C22⋊D511yescharacteristic
598C8—202no—
599C8—202no—
600C5#2347211yescharacteristic, complemented, Sylow
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—202no—
609C22—402no—
610C22—402no—
611C22—402no—
612C22#6552011yescharacteristic
613C22#6552011yescharacteristic
614C22#6552011yescharacteristic
615C22#6552011yescharacteristic
616C22C2×#409211yescharacteristic
617C22C2×#409311yescharacteristic
618C22C2×#409911yescharacteristic
619C4—44no—
620C4—44no—
621C4—44no—
622C4—202no—
623C4—402no—
624C4#6552011yescharacteristic
625C4#6552011yescharacteristic
626C4#6552011yescharacteristic
627C4#6552011yescharacteristic
628C4C2×#409211yescharacteristic
629C4C2×#409211yescharacteristic
630C4C2×#409311yescharacteristic
631C4C2×#409311yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—21no—
637C2—21no—
638C2—21no—
639C2—402no—
640C2C2×#6552011yescharacteristic
subgroupG/H
centerC4C2×#4093
commutator, cosocleC22×C4×C5C24
FrattiniC22×C4C23×D5
FittingC5×#3458C2
socleC2×C5C2×#393
prime radicalC22×#709C22
hypercenter#3458D5
hypocenterC5#23472
LCSG ▷ C22×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C4 ◁ C23×C4 ◁ #3458

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C5×#3458C2semidirect
5. #10602635C2semidirect
6. C2×#64753C2semidirect
7. #10602371C2semidirect
8. C2×#64751C2semidirect
16. C5×#533C22semidirect
32. #64753C22semidirect
33. #64751C22semidirect
34. #64743C22semidirect
48. C2×#709Q8semidirect
50. C22×D20Q8semidirect
52. C22×Dic10Q8semidirect
108. #3458D5semidirect
228. #533C2×D5semidirect
600. C5#23472semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C5×#34582
C2×#652952
C2×#652942
#106026352
C2×#647532
#106023712
C2×#647512
#234725